The cohesive energy of ionic crystals

The cohesive energy of ionic crystals is mainly due to electrostatic interaction and can be calculated on the basis of a point-charge model. Following Born, the cohesive energy (U) of a crystal containing oppositely charged ions with charges Zj and Zj is written as the sum of two terms, one due to attraction and the other due to repulsion ... 

Until the advent of quantum mechanics the reasons for the stability of molecules were unknown. The cohesive energy of ionic crystals could be adequately interpreted on the purely classical basis of the electrical attraction of the oppositely charged ions. Some attempts were made to interpret the interaction of all atoms on the basis of the electrical interaction of positive and negative charges, electrical dipoles, induced dipoles, and so on. These classical calculations indicated that the bonding between two like atoms, such as two hydrogen atoms, should be very much weaker than it is. This is another problem that classical physics failed to solve. 

In the discussions of the kinetic theory of gases and of intermolecular forces, we obtained expressions for properties of matter in bulk in terms of the properties of the individual molecules. In this chapter we will describe the cohesive energy of ionic crystals in terms of the interactions of the ions in the crystals, and some of the properties of metals and covalent crystals in terms of the quantum mechanical picture obtained from the Schrodinger equation. In Chapter 29 we will describe the method for calculating the thermodynamic properties of bulk systems from a knowledge of structure. 

A satisfactory theory of the cohesive energy of ionic crystals can be based almost exclusively on Coulomb s law. If two particles i and j, having charges Zj and Zj, are placed a distance apart in vacuum, the energy of interaction between them is... 

Madelung constant A constant, denoted a, which appears in calculations of the cohesive energy of ionic crystals when the electrostatic interactions between ions in a lattice are summed. The Madelung constant is dimensionless and is a characteristic of the specific crystal structure. It is named for E. Madelung who introduced it in 1918. 

AFMs have also been used to eshmate the cohesive energy of ionic materials with face-centred-cubic structure (Fraxedas et al., 2002a). In these experiments an ultrasharp AFM hp (hp radius / < 10 nm) indents a hat surface of a single crystal and the dynamical mechanical response of the surface during indentahon is transformed into a force plot (applied force vs. penetrahon). It turns out that the... 

It is quite remarkable that electrostatic calculations based on a simple model of integral point charges at the nuclear positions of ionic crystals have produced good agreement with values of the cohesive energy as determined experimentally with use of the Born-Haber cycle. The point-charge model is a purely electrostatic model, which expresses the energy of a crystal relative to the assembly of isolated ions in terms of the Coulombic interactions between the ions. 

In the general theory of ionic crystals (such as table salt, NaCl), a key physical quantity is the cohesive energy Zsxtal of forming the solid crystal from its constituent ions. For sodium chloride, for example, this is the energy lowering in the reaction... 

Although surprisingly circuitous, this expression for the cohesive energy of an ionic crystal carries the full authority of the first law. 

The theorem has the important implication that intramolecular interactions can be calculated by the methods of classical electrostatics if the electronic wave function (or charge distribution) is correctly known. The one instance where it can be applied immediately is in the calculation of cohesive energies in ionic crystals. Taking NaCl as an example, the assumed complete ionization that defines a (Na+Cl-) crystal, also defines the charge distribution and the correct cohesive energy is calculated directly by the Madelung procedure. 

V. Determination of Charges from the Cohesion Energy of an Ionic Crystal... 

The cohesive energy of an ionic crystal is the energy of the crystal relative to the infinitely separated ions, the energy required for the reaction... 

The cohesive energy of a material is a fundamental property and has long been the subject of theoretical and computational approaches. For ionic crystals, quasi-classical approaches have... 

Unfortunately there are no simple theories to predict the cohesive energies of the metals like the coulomb attraction in ionic crystals. More sophisticated quantum mechanical theories using pseudopotential or other modeling techniques are generally required. There are some interesting correlations, however. 

The hard-sphere model, presented above in its most simplified form, was later refined in order to account for the cohesion energy and the elastic properties of ionic crystals. 

Seiler and Dunitz point out that the main reason for the widespread acceptance of the simple ionic model in chemistry and solid-state physics is its ease of application and its remarkable success in calculating cohesive energies of many types of crystals (see chapter 9). They conclude that the fact that it is easier to calculate many properties of solids with integral charges than with atomic charge distributions makes the ionic model more convenient, but it does not necessarily make it correct. 

The parameter is obtained by relating the static dielectric constant to Eg and taking in such crystals to be proportional to a - where a is the lattice constant. Phillips parameters for a few crystals are listed in Table 1.4. Phillips has shown that all crystals with a/ below the critical value of0.785 possess the tetrahedral diamond (or wurtzite) structure when f > 0.785, six-fold coordination (rocksalt structure) is favoured. Pauling s ionicity scale also makes such structural predictions, but Phillips scale is more universal. Accordingly, MgS (f = 0.786) shows a borderline behaviour. Cohesive energies of tetrahedrally coordinated semiconductors have been calculated making use... 

The binding energy of a solid is the energy required to disperse a solid into its constituent atoms, against the forces of cohesion. In the case of ionic crystals, it is given by the Born-Mayer equation. See Crystal. 

A Theoretical Investigation into Some Properties of Ionic Crystals. A Quantum Mechanical Treatment of the Cohesive energy, the Interionic Distance, the Elastic Constants, and the Compression at High Pressures with Numerical Ap-... 

P.-O. Lowdin, Ark. Mat. Astr. Fysik A35, 1 (1947). A Quantum Mechanical Calculation of the Cohesive Energy, the Interionic Distance, and the Elastic Constants of Some Ionic Crystals. 

The London forces are predominant in the adsorption of gases on solid substances such as carbon. In the adsorption on ionic lattices such as salt layers (CaF2 in electric lamps), silicic acid and aluminium oxide the adsorption of the first layer however depends mainly on polarization by electrostatic forces, in which isolated ions, corners and edges will give a larger heat of adsorption than a perfect crystal surface. In adsorption in multimolecular layers the Van der Waals-London energy is, however, predominant as in the cohesion energy. 

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